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Complete asymptotic expansions for the product averges of higher derivetives of Lerch zeta-functions

Masanori Katsurada,
published 2010-07-27

This is a preannouncement version of the forthcoming paper [Ka11]. Let varphi (s,c,lambda) be the Lerch zeta-function defined by (1.1) below, and Im1,m2 (s1, s2; a, lambda) the product average of higher derivatives of varphie (s,x,lambda), given in the form (1.2). The present investigation proceeds with our previous study [Ka2][Ka9] to establish a general explicit formula for (1.2) (Theorem 1); this further leads us to show that a complete asymptotic expansion exists for (1.2) when s1 = sigma + it and s2 = sigma2 - it in the descending order of t as t to pm infty (Theorem 2). The existence of such an asymptotic expansion of (1.2) has been shown in particular when m1 = m2 = 0 and a = 1 by the author [Ka2]; however, it is rather remarkable that a similar asymptotic series still exists in the most general setting into this direction. Our main formula (2.13) with (2.14) and (2.15) is reduced, for e.g., to an improvement upon the previous result (1.6) on the critical line sigma = 1/2 (see Corollary 2.3), and to similar asymptotic expansions of (1.2) in more extended regions (Corollaries 2.1 and 2.2), in particular including the line sigma = 1 (Corollary 2.4).

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